Looking Through the Vortex Glass

Transcription

1 Looking Through the Vortex Glass Lorenz and the Complex Ginzburg-Landau Equation Igor Aronson

2 It started in 1990 Project started in Lorenz Kramer s VW van on the way back from German Alps after unsuccessful attempt to climb mount Watzmanm (2713 m) VW Van similar to that of Lorenz

3 Complex Ginzburg-Landau Equation! = + + " # +! t A A 2 (1 ib ) A (1 ic ) A A A(x,y,z,t) complex amplitude! 2! 2! 2! x 2! y 2! z 2 "= + Laplace + operator b linear dispersion c nonlinear dispersion

4 CGLE Vortex Glass Marc Chagall Glass

5 Vortex Glass Discovered Greg Huber, Preben Alstrom, Tomas Bohr, 1992 Solved on the fastest computer of that time VG called ``frozen state Only black and white images published

6 Understanding Vortex Glass Direct Numerical Solution of the CGLE is not a perfectly good idea all computers choke Need for analytical methods (reduction to discrete equations for particles positions) What is the steady state if any? Influence of effective spiral interaction?

7 Spiral Solution to CGLE A = F( r)exp[ % i! t ± i" + i# ( r) + i$ ] r, " % polar coordiantes, $ = const % phase F( r), # ( r) % unique monotonous functions # ( r) & kr, k % asymptotic wavenumber F(0) = 0, F( ') = 1% k! = c % c % b k % 2 ( ) rotation frequency F(r) Ψ (r) k "! #! exp & $ for 0 2 c b ' c $ b % ( $ )

8 Interaction of Spiral Waves Mostly incorrect results before (and after) Monotonic for small c,b (I.A, Lorenz Kramer, Andreas Weber, 1991) Len Pismen and Alex Nepomnyashii, 1992 Oscillatory with Symmetry Breaking, I.A., L.K., A.W., 1993 Core Instability, I.A., L.K. A.W I.A & L.K. Review Mod. Phys 2002

9 Phase diagram of 2D CGLE Vortex Liquid Vortex Glass Amplitude (or defect) Turbulence OR-oscillatory range (symmetry breaking) EI Kramer-Eckhaus instability for spirals BF Newell-Benjamin-Fair limit 1+bc=0 AI- absolute instability for spirals and transition to chaos

10 Vortex Glass and Eckhaus Instability Nucleation of big spirals after very long transient

11 Vortex Glass in 3D Core Unstable Vortices in Oscillatory range show length oscillations

12 Mythology of Spiral interaction Exponential decay of interaction Monotonic range c<0.845 & b=0: -weak repulsion irrespectively of charge -no symmetry breaking Oscillatory range c>0.845 & b=0: -oscillatory interaction vs distance -meta-stable bound states -symmetry breaking v velocity of the spiral core ϕ -phase of the spiral X -distance to the shock line y shock line v v -X 0 X Biktashev, 1989 Aranson, Kramer & Weber, Phys Rev. E, 1991,1993 Pismen and Nepomnyashii 1991

13 Monotonic case: pair interaction ' rj r k / 2, B 0.48 (" ) screening length l ~ 1/ ck ~ exp 2c $ % for c $ 0 # 2 ck X # 2 ck X 3 ' e 2 e vn = # 2 c " k B ; v! = 2 mc " k ; " ck X " ck X X = # & equations for "complex" position z = x + iy. # 2 ck X 2 ' j k z j = 2 c " k ( cb + imk ) r j # r k " r # r e ck X Oblique interaction Oppositely charged spirals repel and drift Likely charged spirals repel and rotate Bypassing ultaslow time scale T~1/(c 3 k 2 )!!! Real GLE singular limit to CGLE

14 Oscillatory Case: Asymptotic Method Spiral pair spiral with the boundary v y (! ) " # $ % 0 A = F( r) + exp[ i t + i + i ( r) + i ( t)] r & r ' r ( t), v = r! ( t) ' spiral core velocity 0 0! = C r exp[ ' pr + i# ] + C r exp[ pr + i# ] µ µ 1 2 ' linearized stationary solution -X x Constants C determined from b.c v(t) and ϕ 0 (t) determined from numerical orthogonalization at the core I.A, Weber and Kramer, 1993

15 Examples of Bound States Oppositely charged- drift Likely charged-rotate c=1.5,b=0

16 Symmetry Breaking Big Brother always wins!!!

17 Final State after Symmetry Breaking Spiral defect sink

18 Velocity of the Shock Line ω 1 & k V 1 ω 2 & k 2! "! 1 ( )( ) ( ) 1 2 V = = b " c k1 + k2 = Vg1 + Vg 2 k1 " k2 2 Position of the Shock Line Spiral Solution Full Phase A = F( r )exp[% i" t + im # + i$ ( r ) + i! ] 1,2 1,2 1, 2 1,2 1,2 #( X ) =! ( r) + " $ kx + " " X = k( X # X ) +! #! Displacement of the shock

19 Frequency of Spiral in Finite Domain

20 Oscillatory Case: Equations of Motion & 2 % px % k 1% k e ' % µ vn = Im X Im( Cx / Cy ) ( " Cy 2# px ) * + & 2 % px % k 1% k e ' % µ v! = Re X % vn Re( Cx / Cy ) ( " Cy 2# px ) * + & ' % µ = Im X Im( C10 / C00) dt ( " C 2# px ) 2 % px d$ % k 1% k e X * 00 + = r % r / 2 + ( $ % $ ) / 2k jk j k j k Self-Consistently determined velocity, phases, and positions of shocks

21 Multivortex states Monotonic range: vortex liquid, chaotic motion even for (b-c) 0 Oscillatory range: (c-b)/(1+bc)>0.85, symmetry breaking and vortex glass, ultraslow motion, intermittency, ageing Carolina Brito, Hugues Chate, I.A, 2003

22 Numerical Simulations CGLE, periodic boundary conditions, random initial conditions, up to 4096x4096 length units, up to 10 7 time units, b=0, c= ODEs for spiral positions, periodic boundary conditions, random initial spiral positions, both oscillatory and monotonic cases, # of spirals: , time units, spiral parameters taken from theory Two cases: -fixed number of spirals -deterministic annihilation rule of close spirals (after CGLE dynamics)

23 Monotonic Case Vortex Liquid Monotonic case: (c-b)/(1+bc)<0.85 -relaxation to equilibrium spiral density -normal diffusion of individual spirals -well-defined effective temperature -almost Maxwellian spiral velocity distribution Reduced Equations 2D CGLE

24 Comparison with CGLE: Quantitative agreement

25 Normal Diffusion. Why? EoM are not variational if total charge is zero (e.g. periodic bc). " X r r. " X ' j " k e 1 e z j = ( cb + imk ) ;! j = r " r X 2c X j k However, likely-charged spirals obey variational principle:. ' z j = "( cb + i)! F! z Likely-charged spirals tend to form hexagonal lattice Likely-charged spirals tend to crystallize in the liquid (short-range order in liquid) * j

26 Monotonic Case. Instability of Lattice r " r ' e 1 z cb im r " r X 2c. " X. " X j k j = ( + k ) ;! j = X = r " r / 2 + (! "! ) j k j k j + positive spirals, * negative spirals, ο shocks k e X

27 Instability of Lattice $ z % & ' # ( jk ) jk t i j i k! x y e! + " + " Positions of nodal lines λ(κ)=0! +! # 2X!! = 0,!! x y x y Perturbation to square lattice λ(κ) growth rate, κ wavevector y! =! sin ";! =! cos" x y θ x 2 sin 2 " = 2 X critical angle

28 Vortex Glass-Oscillatory Case Oscillatory case: (c-b)/(1+bc)<0.85 -slow logarithmic decay of spiral density aging -intermittency and activity bursts -no well-defined effective temperature Reduced equations 2D CGLE

29 Comparison with 2D CGLE

30 Measured Quantities number of Spirals N temperature (activity) T T = 1 " A ## dxdy L L " t x y S continuum case T spir 1 dr 1 d! j = $ = $ N dt N dt N N j ; Tshock j= 1 j= 1 discrete case

31 Formation of Vortex Glass c=1.2,b=0 L=120, N=26 t=0 t=300 t=82,000

32 2D CGLe & Vortex Simulations

33 Intermittency of Vortex Glass (small system size/low density) CGLE Vortex Simulations T spir T shock

34 Large-Scale Simulations Intermittent activity, slow decay of spiral number

35 Vortex glass quantized vortex domain sizes Vortex Size Distribution (CGLE)

36 Mechanism of Frustration in Vortex Glass EoM in oscillatory case do not have stationary solution Points where v=0 & dϕ/dt=0 do not coincide in general spirals & 2 % px % k 1% k e ' % µ vn = Im X Im( Cx / Cy ) ( " Cy 2# px ) * + & 2 % px % k 1% k e ' % µ v! = Re X % vn Re( Cx / Cy ) ( " Cy 2# px ) * +. & 2 % px % k 1% k e ' % µ $ = Im X Im( C10 / C00) ( " C 2# px ) * 00 + v=0 dϕ/dt=0

37 ODE simulations without annihilation at low density but large-size limit: averaging out of localized intermittent activity:

38 liquid state at high density, even in oscillatory regime recover vortex liquid not observed (not observable?) in PDE

39 Phase diagram without annihilation ρ-average spiral density

40 With annihilation: ageing- degradation of VG N! 1 log t

41 Conclusions The CGLE chokes the fastest computers Simple system totally unexpected behavior In ultra-slow time scale the dynamics is highly chaotic Statistical features the vortex glass are still not totally clear Vortex glass is still not shattered into small bits